Lloc1-convergence of Jacobians of Sobolev homeomorphisms via area formula

Abstract

We prove that given a sequence of homeomorphisms $f_k:\Omega \to R^n$ convergent in $W^{1,p}(\Omega ,R^n)$, $p\ge 1$ for $n=2$ and $p>n-1$ for $n\ge 3$, to a homeomorphism f which maps sets of measure zero onto sets of measure zero, Jacobians $J{f}_k$ converge to Jf in $L_{loc}^1\left(\Omega \right)$. We prove it via Federer's area formula and investigation of when $\left|f_k\right(E\left)\right|\to \left|f\right(E\left)\right|$ as $k\to \infty$ for Borel subsets $E⋐\Omega$.